and \(\Phi = K + \lambda I_n\) , with \(K\)  the \(n \times n\)  matrix with \(ij\) th element \(\kappa (\mathbf {x}_i,\mathbf {x}_j; \rho )\) . The values of \(\lambda \)  and \(\rho \)  can be determined using using generalised cross validation [1]}. Under certain regularity conditions, the most stringent being that \(\mathbf {x}_1, \dots , \mathbf {x}_n\)  are an independent and identically distributed sample from \(U(\mathcal {X})\) , [1]} show that \(\hat{\theta }_{TW}\)  is a consistent estimator of \(\theta _{L^2}\) . Moreover, \(\hat{\theta }_{TW}\)  has the following asymptotic normal distribution
\(\hat{\theta }_{TW}\sim \mathrm {N}\left(\theta _{L^2}, V^{-1} WV^{-1} \right),\) 
